Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12,15,20,16,21,31,18)(13,22,25,33,26,36,23,17)(14,27,28,35,30,34,29,19)(24,32,38,41,40,42,39,37);(4,7)(8,9)(11,13,16,26)(12,17,21,33)(14,24,30,40)(15,23,31,25)(18,22,20,36)(19,32,35,42)(27,37,34,41)(28,39,29,38);(12,18)(13,23)(15,31)(19,27)(20,21)(22,36)(24,38)(25,26)(28,29)(34,35)(37,41)(39,40);(3,5,6,10)(4,8,7,9)(11,14,16,30)(12,19,21,35)(13,24,26,40)(15,29,31,28)(17,32,33,42)(18,27,20,34)(22,37,36,41)(23,38,25,39)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346925);